Graph Invariants (covering design) I am not a mathematician so please take that into consideration when formulating your answers. 
Technically 2 graphs are NOT isomorphic if any one of the countless graph invariants (i.e. vertices, edges, etc…) are not the same for both graphs.
I would like to know in this case which graph invariant is different between Graph A and Graph B for this covering design (v=10, k=6, t=3) which proves that Graph A and Graph B are NOT isomorphic. Furthermore, how is it calculated?
Graph A
1,  2,  3,  4,  6,  7
1,  2,  3,  5,  7, 10
1,  2,  3,  8,  9, 10
1,  2,  4,  6,  8, 10
1,  3,  4,  5,  6,  9
1,  4,  5,  7,  8,  9
2,  4,  5,  6,  9, 10
2,  5,  6,  7,  8,  9
3,  4,  5,  7,  8, 10
3,  6,  7,  8,  9, 10
Graph B
1,  2,  3,  4,  6,  7
1,  2,  3,  5,  8, 10
1,  2,  3,  7,  9, 10
1,  2,  4,  6,  8, 10
1,  3,  4,  5,  6,  9
1,  4,  5,  7,  8,  9
2,  4,  5,  6,  9, 10
2,  5,  6,  7,  8,  9
3,  4,  5,  7,  8, 10
3,  6,  7,  8,  9, 10
The only difference between Graph A and Graph B is in blocks 2 & 3 where the 7 and the 8 are inverted. All the other blocks are the same.
Thanks
Roy
 A: In graph-theoretic terms covering designs $(v,k,t)$ represent a specific form of this more general case: 
Select a graph $G=(V_1,E_1)$ and a second graph $H=(V_2,E_2)$ with $|V_1| \geq |V_2|$ and $|E_1| \geq |E_2|$. 
Then the task is to select subgraph(s) $g_i$ of $G$ isomorphic to $H$ such that a set of these subgraphs $(g_1,g_2,...g_n)$ covers some predefined features of $G$ ie. all edges, $n$-cycles, or $n$-cliques.
Usually the specific case refernced by the OP with $(v,k,t)$ given corresponds to this specific instance:
$G=K_v$, select a set of $k$-cliques in $G$ that covers every combination of $t$ nodes in $G$ at least once. Most frequently the problem is analyzed with $t=2$ which gives a set of $k$-cliques in $K_v$ that covers each edge of $K_v$. In the OP's case he has provide two sets of $6$-cliques, which, taken over an arbitrary labeling of the vertices of $K_{10}$ from $(1,2,...10)$, cover all $2$-paths ($3$-cycles) in $K_{10}$.
This notation still does not provide a definitive answer to the original question because there are many ways the individual sets of blocks could be combined into a "graph", but hopefully it provides some motivation for the notation and the question.
